The process matrix formalism formalises non-classical causal structures, which are expected to emerge in a theory that combines quantum indeterminacy with the dynamical causal structure of general relativity. The framework relies on the notion of local laboratoriesâ an abstraction of spacetime events or regionsâ where local operations and measurements can be performed. In a theory of quantum gravity, we expect that it should not be possible to label laboratories a priori: this requires a background reference frame (although possibly non-classical), while general relativity is background independent. I will present a background independent formulation of the process matrix formalism that incorporates a discrete version of background independence: invariance under permutations. Although the formalism abandons an absolute notion of â local operationâ , such a notion re-emerges as relative to a physical reference frame. Permutation symmetry also implies some type of superselection rule, although, for quantum processes, this cannot be interpreted as implying the existence of a conserved charge with a definite value. Finally, I will comment on the possibility to extend the approach to include quantum coordinate transformations.